Group #50318653 — subgroups

Subgroups — C₁₅₁×#13728 #50318653

Order 38656.

238 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SD₆₄×C₁₅₁	C₂	1	2	yes	maximal, complemented
3	C₂×D₃₂×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#2249×C₁₅₁	C₂	1	2	yes	maximal, complemented
5	C₄×C₃₂×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₃₂×C₁₅₁	C₂²	1	2	yes	—
7	C₂×SD₃₂×C₁₅₁	—	2	2	no	—
8	C₂×D₁₆×C₁₅₁	C₂²	1	2	yes	—
9	#495×C₁₅₁	—	2	2	no	—
10	Q₆₄×C₁₅₁	—	2	2	no	—
11	SD₆₄×C₁₅₁	—	2	4	no	—
12	D₃₂×C₁₅₁	—	2	2	no	—
13	C₂×C₃₂×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₂×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₅₁	—	2	2	no	—
17	C₂×SD₁₆×C₁₅₁	—	4	2	no	—
18	C₂×D₈×C₁₅₁	—	2	2	no	—
19	#124×C₁₅₁	—	4	2	no	—
20	Q₃₂×C₁₅₁	—	2	2	no	—
21	Q₃₂×C₁₅₁	—	2	2	no	—
22	SD₃₂×C₁₅₁	—	4	4	no	—
23	D₁₆×C₁₅₁	—	2	2	no	—
24	D₁₆×C₁₅₁	—	2	2	no	—
25	C₂×C₁₆×C₁₅₁	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₁₅₁	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
28	C₃₂×C₁₅₁	D₄	1	2	yes	—
29	C₃₂×C₁₅₁	D₄	1	2	yes	—
30	C₂×Q₈×C₁₅₁	—	4	2	no	—
31	C₂×D₄×C₁₅₁	—	4	2	no	—
32	Q₁₆×C₁₅₁	—	4	2	no	—
33	Q₁₆×C₁₅₁	—	4	2	no	—
34	SD₁₆×C₁₅₁	—	8	4	no	—
35	D₈×C₁₅₁	—	4	2	no	—
36	D₈×C₁₅₁	—	4	2	no	—
37	C₂×C₈×C₁₅₁	D₈	1	1	yes	characteristic
38	C₂×C₈×C₁₅₁	C₂×D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₁₅₁	—	8	4	no	—
40	C₄²×C₁₅₁	D₈	1	1	yes	characteristic
41	C₁₆×C₁₅₁	D₈	1	2	yes	—
42	C₁₆×C₁₅₁	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₁₅₁	C₂×D₄	1	1	yes	characteristic
44	C₂³×C₁₅₁	—	8	2	no	—
45	Q₈×C₁₅₁	—	8	2	no	—
46	Q₈×C₁₅₁	—	8	2	no	—
47	D₄×C₁₅₁	—	8	2	no	—
48	D₄×C₁₅₁	—	8	2	no	—
49	C₂×C₄×C₁₅₁	—	8	2	no	—
50	C₂×C₄×C₁₅₁	D₁₆	1	2	yes	—
51	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
52	C₈×C₁₅₁	D₁₆	1	2	yes	—
53	C₈×C₁₅₁	#127	1	1	yes	characteristic
54	C₈×C₁₅₁	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₁₅₁	—	16	2	no	—
56	C₂²×C₁₅₁	—	16	4	no	—
57	C₂²×C₁₅₁	C₂×D₁₆	1	1	yes	characteristic
58	C₄×C₁₅₁	—	2	2	no	—
59	C₄×C₁₅₁	—	16	2	no	—
60	C₄×C₁₅₁	#493	1	1	yes	characteristic
61	C₄×C₁₅₁	C₂×D₁₆	1	1	yes	characteristic
62	C₂×C₁₅₁	—	32	2	no	—
63	C₂×C₁₅₁	#2246	1	1	yes	characteristic
64	C₂×C₁₅₁	#2262	1	2	yes	—
65	#13728	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₁₅₁	#13728	1	1	yes	characteristic, direct product factor, Sylow
67	C₂×Q₆₄	C₂×C₁₅₁	1	1	yes	characteristic, complemented
68	C₂×SD₆₄	C₂×C₁₅₁	1	2	yes	complemented
69	C₂×D₃₂	C₂×C₁₅₁	1	1	yes	characteristic
70	#2249	C₂×C₁₅₁	1	2	yes	complemented
71	C₄×C₃₂	C₂×C₁₅₁	1	1	yes	characteristic, complemented
72	C₂×Q₃₂	C₂²×C₁₅₁	1	2	yes	—
73	C₂×SD₃₂	—	2	2	no	—
74	C₂×D₁₆	C₂²×C₁₅₁	1	2	yes	—
75	#495	—	2	2	no	—
76	Q₆₄	—	2	2	no	—
77	SD₆₄	—	2	4	no	—
78	D₃₂	—	2	2	no	—
79	C₂×C₃₂	C₂²×C₁₅₁	1	1	yes	characteristic
80	C₂×C₃₂	C₂²×C₁₅₁	1	1	yes	characteristic
81	C₄×C₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×SD₁₆	—	4	2	no	—
84	C₂×D₈	—	2	2	no	—
85	#124	—	4	2	no	—
86	Q₃₂	—	2	2	no	—
87	Q₃₂	—	2	2	no	—
88	SD₃₂	—	4	4	no	—
89	D₁₆	—	2	2	no	—
90	D₁₆	—	2	2	no	—
91	C₂×C₁₆	D₄×C₁₅₁	1	1	yes	characteristic
92	C₂×C₁₆	C₂³×C₁₅₁	1	1	yes	characteristic
93	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
94	C₃₂	D₄×C₁₅₁	1	2	yes	—
95	C₃₂	D₄×C₁₅₁	1	2	yes	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×D₄	—	4	2	no	—
98	Q₁₆	—	4	2	no	—
99	Q₁₆	—	4	2	no	—
100	SD₁₆	—	8	4	no	—
101	D₈	—	4	2	no	—
102	D₈	—	4	2	no	—
103	C₂×C₈	D₈×C₁₅₁	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄×C₁₅₁	1	1	yes	characteristic
105	C₂²⋊C₄	—	8	4	no	—
106	C₄²	D₈×C₁₅₁	1	1	yes	characteristic
107	C₁₆	D₈×C₁₅₁	1	2	yes	—
108	C₁₆	C₂×D₄×C₁₅₁	1	1	yes	characteristic
109	C₁₆	C₂×D₄×C₁₅₁	1	1	yes	characteristic
110	C₂³	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	Q₈	—	8	2	no	—
113	D₄	—	8	2	no	—
114	D₄	—	8	2	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	D₁₆×C₁₅₁	1	2	yes	—
117	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
118	C₈	D₁₆×C₁₅₁	1	2	yes	—
119	C₈	#127×C₁₅₁	1	1	yes	characteristic
120	C₈	C₂×D₈×C₁₅₁	1	1	yes	characteristic
121	C₂²	—	16	2	no	—
122	C₂²	—	16	4	no	—
123	C₂²	C₂×D₁₆×C₁₅₁	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	16	2	no	—
126	C₄	#493×C₁₅₁	1	1	yes	characteristic
127	C₄	C₂×D₁₆×C₁₅₁	1	1	yes	characteristic
128	C₂	—	32	2	no	—
129	C₂	#2246×C₁₅₁	1	1	yes	characteristic
130	C₂	#2262×C₁₅₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₆₄×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×SD₆₄×C₁₅₁

C₂

yes

maximal, complemented

C₂×D₃₂×C₁₅₁

C₂

yes

maximal, characteristic

#2249×C₁₅₁

C₂

yes

maximal, complemented

C₄×C₃₂×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₃₂×C₁₅₁

C₂²

yes

—

C₂×SD₃₂×C₁₅₁

—

C₂×D₁₆×C₁₅₁

C₂²

yes

—

#495×C₁₅₁

—

Q₆₄×C₁₅₁

—

SD₆₄×C₁₅₁

—

D₃₂×C₁₅₁

—

C₂×C₃₂×C₁₅₁

C₂²

yes

characteristic

C₂×C₃₂×C₁₅₁

C₂²

yes

characteristic

C₄×C₁₆×C₁₅₁

C₂²

yes

characteristic

C₂×Q₁₆×C₁₅₁

—

C₂×SD₁₆×C₁₅₁

—

C₂×D₈×C₁₅₁

—

#124×C₁₅₁

—

Q₃₂×C₁₅₁

—

Q₃₂×C₁₅₁

—

SD₃₂×C₁₅₁

—

D₁₆×C₁₅₁

—

D₁₆×C₁₅₁

—

C₂×C₁₆×C₁₅₁

D₄

yes

characteristic

C₂×C₁₆×C₁₅₁

C₂³

yes

characteristic

C₄×C₈×C₁₅₁

D₄

yes

characteristic

C₃₂×C₁₅₁

D₄

yes

—

C₃₂×C₁₅₁

D₄

yes

—

C₂×Q₈×C₁₅₁

—

C₂×D₄×C₁₅₁

—

Q₁₆×C₁₅₁

—

Q₁₆×C₁₅₁

—

SD₁₆×C₁₅₁

—

D₈×C₁₅₁

—

D₈×C₁₅₁

—

C₂×C₈×C₁₅₁

D₈

yes

characteristic

C₂×C₈×C₁₅₁

C₂×D₄

yes

characteristic

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

D₈

yes

characteristic

C₁₆×C₁₅₁

D₈

yes

—

C₁₆×C₁₅₁

C₂×D₄

yes

characteristic

C₁₆×C₁₅₁

C₂×D₄

yes

characteristic

C₂³×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

D₁₆

yes

—

C₂×C₄×C₁₅₁

C₂×D₈

yes

characteristic

C₈×C₁₅₁

D₁₆

yes

—

C₈×C₁₅₁

#127

yes

characteristic

C₈×C₁₅₁

C₂×D₈

yes

characteristic

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂×D₁₆

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#493

yes

characteristic

C₄×C₁₅₁

C₂×D₁₆

yes

characteristic

C₂×C₁₅₁

—

C₂×C₁₅₁

#2246

yes

characteristic

C₂×C₁₅₁

#2262

yes

—

#13728

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#13728

yes

characteristic, direct product factor, Sylow

C₂×Q₆₄

C₂×C₁₅₁

yes

characteristic, complemented

C₂×SD₆₄

C₂×C₁₅₁

yes

complemented

C₂×D₃₂

C₂×C₁₅₁

yes

characteristic

#2249

C₂×C₁₅₁

yes

complemented

C₄×C₃₂

C₂×C₁₅₁

yes

characteristic, complemented

C₂×Q₃₂

C₂²×C₁₅₁

yes

—

C₂×SD₃₂

—

C₂×D₁₆

C₂²×C₁₅₁

yes

—

#495

—

Q₆₄

—

SD₆₄

—

D₃₂

—

C₂×C₃₂

C₂²×C₁₅₁

yes

characteristic

C₂×C₃₂

C₂²×C₁₅₁

yes

characteristic

C₄×C₁₆

C₂²×C₁₅₁

yes

characteristic

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

#124

—

Q₃₂

—

Q₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

C₂×C₁₆

D₄×C₁₅₁

yes

characteristic

C₂×C₁₆

C₂³×C₁₅₁

yes

characteristic

C₄×C₈

D₄×C₁₅₁

yes

characteristic

C₃₂

D₄×C₁₅₁

yes

—

C₃₂

D₄×C₁₅₁

yes

—

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

100

SD₁₆

—

101

D₈

—

102

D₈

—

103

C₂×C₈

D₈×C₁₅₁

yes

characteristic

104

C₂×C₈

C₂×D₄×C₁₅₁

yes

characteristic

105

C₂²⋊C₄

—

106

C₄²

D₈×C₁₅₁

yes

characteristic

107

C₁₆

D₈×C₁₅₁

yes

—

108

C₁₆

C₂×D₄×C₁₅₁

yes

characteristic

109

C₁₆

C₂×D₄×C₁₅₁

yes

characteristic

110

C₂³

—

111

Q₈

—

112

Q₈

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

D₁₆×C₁₅₁

yes

—

117

C₂×C₄

C₂×D₈×C₁₅₁

yes

characteristic

118

C₈

D₁₆×C₁₅₁

yes

—

119

C₈

#127×C₁₅₁

yes

characteristic

120

C₈

C₂×D₈×C₁₅₁

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

C₂×D₁₆×C₁₅₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

#493×C₁₅₁

yes

characteristic

127

C₄

C₂×D₁₆×C₁₅₁

yes

characteristic

128

C₂

—

129

C₂

#2246×C₁₅₁

yes

characteristic

130

C₂

#2262×C₁₅₁

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₅₁	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₅₁
prime radical	C₂×D₃₂×C₁₅₁	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₄²×C₁₅₁ ◁ C₄×C₈×C₁₅₁ ◁ C₄×C₁₆×C₁₅₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₅₁

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₁₅₁

prime radical

C₂×D₃₂×C₁₅₁

C₂

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₅₁ ◁ C₄²×C₁₅₁ ◁ C₄×C₈×C₁₅₁ ◁ C₄×C₁₆×C₁₅₁ ◁ G

N	H ≅ G/N	type
1. C₂×Q₆₄×C₁₅₁	C₂	semidirect
2. C₂×SD₆₄×C₁₅₁	C₂	semidirect
4. #2249×C₁₅₁	C₂	semidirect
5. C₄×C₃₂×C₁₅₁	C₂	semidirect
65. #13728	C₁₅₁	direct (N × H)
66. C₁₅₁	#13728	direct (N × H)
67. C₂×Q₆₄	C₂×C₁₅₁	semidirect
68. C₂×SD₆₄	C₂×C₁₅₁	semidirect
70. #2249	C₂×C₁₅₁	semidirect
71. C₄×C₃₂	C₂×C₁₅₁	semidirect

H ≅ G/N

type

1. C₂×Q₆₄×C₁₅₁

C₂

semidirect

2. C₂×SD₆₄×C₁₅₁

C₂

semidirect

4. #2249×C₁₅₁

C₂

semidirect

5. C₄×C₃₂×C₁₅₁

C₂

semidirect

65. #13728

C₁₅₁

direct (N × H)

66. C₁₅₁

#13728

direct (N × H)

67. C₂×Q₆₄

C₂×C₁₅₁

semidirect

68. C₂×SD₆₄

C₂×C₁₅₁

semidirect

70. #2249

C₂×C₁₅₁

semidirect

71. C₄×C₃₂

C₂×C₁₅₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Q₆₄×C₁₅₁	2
C₂×SD₆₄×C₁₅₁	2
C₂×D₃₂×C₁₅₁	2
#2249×C₁₅₁	2
C₄×C₃₂×C₁₅₁	2
#13728	151

subgroup

index

C₂×Q₆₄×C₁₅₁

C₂×SD₆₄×C₁₅₁

C₂×D₃₂×C₁₅₁

#2249×C₁₅₁

C₄×C₃₂×C₁₅₁

#13728

151

Subgroups — C₁₅₁×#13728 #50318653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#13728 #50318653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#13728 #50318653